Research output: Contribution to journal › Article › peer-review
Pronormality of Hall Subgroups in Their Normal Closure. / Vdovin, E. P.; Nesterov, M. N.; Revin, D. O.
In: Algebra and Logic, Vol. 56, No. 6, 01.01.2018, p. 451-457.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Pronormality of Hall Subgroups in Their Normal Closure
AU - Vdovin, E. P.
AU - Nesterov, M. N.
AU - Revin, D. O.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent.
AB - It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent.
KW - normal closure
KW - pronormal subgroup
KW - π-Hall subgroup
KW - pi-Hall subgroup
UR - http://www.scopus.com/inward/record.url?scp=85042436357&partnerID=8YFLogxK
U2 - 10.1007/s10469-018-9467-8
DO - 10.1007/s10469-018-9467-8
M3 - Article
AN - SCOPUS:85042436357
VL - 56
SP - 451
EP - 457
JO - Algebra and Logic
JF - Algebra and Logic
SN - 0002-5232
IS - 6
ER -
ID: 10428358